USA vacations/trips where you do not need a car?

New Orleans has good public transportation, there are lots of taxis, Lyft & Uber are active, and the city is very bicycle-friendly.

www.neworleansonline.com 

Chicago has fabulous public transportation. All around the downtown area. And you can take the Metra trains around the city and out into the burbs. You can take the Amtrak train to Milwaukee for the day or over night.

Chicago  has about anything you could want to do.

San Francisco, Seattle or Honolulu are all doable without a car.

Another response mentioned Key West, which I was going to suggest.  IT's very walkable, no hills, and if you get tired, you can uber or take a pedicab.  That said, hotels are pretty expensive.  

Ditto for San Franscisco.  It's got a great public transportation system.  So does Vancouver, if you want to go a little bit out of the country.  

Depending on where exactly you stay in these cities, you can find pretty good public transportation to get you around - Los Angeles, Seattle, Atlanta, San Francisco.  

There are other places that are largely walkable if you stay in the right location - Las Vegas (also has some degree of accessible public transportation), New Orleans, Miami, Charleston SC, Savannah GA, Myrtle Beach (which I hate).

You can also visit theme park resorts such as Disney World, Disneyland, and Universal Orlando without a car. 

San Francisco has great public transportation. New Orleans is walkable if you stay near the French Quarter. Atlanta has subway/train if you stay near downtown. Seattle has good public transportation and is walkable. 

Lots, but what do you enjoy doing?  Nantucket doesn’t need a car.  Hilton Head can be done without one depending on which area, Key West, Charleston, Sanibel island, mount Desert Maine, etc. if you stay in the environs many places don’t need cars. And most have Uber too. There is always the hop on hop off buses too.